Consider the closed-loop response during the dynamic distillation of an ideal binary mixture in the column shown in Fig. 13-107, under two assumptions of constant relative volatility at a value of 2.0 and constant molar vapor flow for a saturated liquid feed to tray Ns. Following the development by Luyben (op. cit.), it is not necessary to include energy-balance equations for each tray or to treat temperature and pressure as variables. Overhead vapor leaving top tray NT is totally condensed for negligible liquid holdup with condensate flowing to a reflux drum having constant and perfectly mixed molar liquid holdup Md. The reflux rate LNt+1 is varied by a proportional-integral (PI) feedback controller to control distillate composition for a set point of 0.98 for the mole fraction xD of the light component. Holdup of reflux in the line leading back to the top tray is neglected. Under dynamic conditions, yNr may not equal xD.
At the bottom of the column, a liquid sump of constant and perfectly mixed molar liquid holdup MB is provided. A portion of the liquid flowing from this sump passes to a thermosiphon reboiler, with the
remainder taken as bottoms product at a molar flow rate B. Vapor boil-up generated in the reboiler is varied by a PI feedback controller to control bottoms' composition with a set point of 0.02 for the mole fraction xB of the light component. Liquid holdups in the reboiler and lines leading from the sump are assumed to be negligible. The composition of the boil-up yB is assumed to be in equilibrium with xB.
The liquid holdup Mn on each of the NT equilibrium trays is assumed to be perfectly mixed but will vary as liquid rates leaving the trays vary. Vapor holdup is assumed to be negligible everywhere. Tray molar vapor rates V vary with time but at any instant in time are everywhere equal.
The dynamic material-balance and phase equilibrium equations corresponding to this description are as follows:
Where Fn is nonzero only for tray Ns, y and x refer to the light component only such that the corresponding mole fractions for the heavy component are (1 - y) and (1 - x), Ln and Mn are the initial steady-state values, and P is a constant that depends on tray hydraulics.
For the condenser-reflux-drum combination:
For the reboiler:
1 + (a - 1)xB The two PI-controller equations are
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